When Chicago Shakespeare Theater staged “The Taming of the Shrew” in 2010, playwright Neil LaBute wrote a new frame for British director Josie Rourke that involved a tempestuous lesbian love triangle and did nothing whatsoever to enhance Shakespeare’s play.

CST artistic director Barbara Gaines, who conceived and directed the current production, hasn’t made the same mistake. While she’s kept the notion of “Shrew” as a play-within-a-play, she’s had Second City’s Ron West create a frame that complements and counterpoints the Bard’s original.

The story is set in Chicago in 1919, amid demonstrations on the eve of the vote for woman’s suffrage, and the cast is made up entirely of women, a canny twist on men playing all the female roles in Elizabethan times. They are the ladies of the Columbia Women’s Club—a fictional take on clubs that were very popular at the time—and they’ve gathered for a final rehearsal of “Shrew” that’s beset by storm-related technical problems that have relegated them to the club’s parlor (sumptuously designed by Kevin Depinet) with little in the way of scenery or costumes (wittily contrived by Susan E. Mickey).

The heart of Gaines’ concept is that each woman has a distinctive personality that’s affected by the Shakespearean role she’s playing, her colleagues, and the marches right outside their door. They start out with diverse opinions about whether or not women should get the vote—and their roles in relationship to their husbands (virtually all of them are married)–but by the end, they are all in favor of the 19th Amendment.

This is a lot for the audience to absorb and keep straight, but the conceit works because the acting is generally excellent, and most of the actors are equally good at portraying the club women and Shakespeare’s characters. It helps that West’s sections of the script are very clever, often funny, and full of topical references that were relevant a century ago and are as apropos today.

Mrs. Louise Harrison as Katherine (Alexandra Henrikson) arguably undergoes the most radical transformation. She starts out being anti-suffrage and happy to let her husband take the lead, in other words the opposite of the shrewish Kate, the older daughter wealthy Padua merchant Baptista insists on marrying off before he’ll let one of many suitors wed his demure younger daughter, Bianca. By the finale, Louise has been caught up in the fray outside and outraged by the treatment of women, while Kate, though not exactly tamed by Petruchio, has formed a workable partnership with him.

Admittedly, Gaines seems to have trimmed and softened Petruchio’s abuse of his reluctant wife, but what’s especially interesting about Crystal Lucas-Perry who plays Mrs. Victoria Van Dyne playing him is that she’s one of the few who comes close to being convincing as a man. That adds complexity and dimension to a character who can be simultaneously annoying and smart.

Of the many mini dramas swirling throughout the evening, Mrs. Emily Ingersoll playing the obedient Bianca (Olivia Washington) finds the courage to stand up to her mother, Mrs. Mildred Sherman (Rita Behn), president of the club, who portrays old Grumio and the Widow, in between bossing around her senator husband on the phone and telling him how to vote. Emily’s eventual outspokenness is a nice variation on the behavior of her Bianca, who gets her way by being devious with the suitor she favors, Lucentio, played by Miss Olivia Twist (Kate Marie Smith), a late replacement and newcomer to the club who gradually finds acceptance.

Bianca’s other suitors include Mrs. Beatrice Welles (a tribute to Orson Welles’ mother, the program tells us) as Hortensio (Tina Gluschenko), whose idea it is to have his friend Petruchio woo Kate; Miss Judith Smith as Gremio (Hollis Resnik), and Mrs. Dorothy Mercer as Tranio (Heidi Kettenring) who, disguised as his master, Lucentio, convinces Dr. Fannie Emmanuel’s Baptista (E. Faye Butler) that he’s the right match. Mrs. Mercer also is the director of “Shrew,” and Miss Smith is responsible for the scenery, such as it is (Resnik has a ball).

In all honesty, the interactions, arguments, and antics of the club women threaten to overshadow the Shakespeare at times, but it’s all a lot fun—with songs of the era thrown in for good measure—and “The Taming of the Shrew” gets happily transformed rather than lost.